Constructive recognition of classical groups in their natural representation.
type: Item
instance of: scholarly article
title: Constructive recognition of classical groups in their natural representation.
zbMATH Open document ID: 1054.20001
DOI: 10.1016/S0747-7171(02)00132-3
author: Peter A. Brooksbank
published in: Journal of Symbolic Computation
publication date: 2004-03-14T00:00:00Z
review text: The author gives a new algorithm to recognize constructively classical groups (SP, SU, \(\Omega^\varepsilon\)) in their natural representation. (See \textit{W. M. Kantor} and \textit{Á. Seress} [Groups, combinatorics and geometry. Proceedings of the L.M.S. Durham symposium, Durham, UK, July 16-26, 2001. River Edge, NJ: World Scientific. 123-137 (2003; Zbl 1052.20001)] for a description of the context.) The algorithm is ``Las Vegas'' in that it uses random elements, but verifies the result to be correct. The heart of the construction is to find subgroups of type \(\Omega^+(4,q)\), respectively \(\text{Sp}(4,q)\) or \(\text{SU}(4,q)\), these then are used to span the whole group. A complete complexity analysis is given that shows the algorithm to perform notably better than the black-box algorithm of \textit{W. M. Kantor} and \textit{Á. Seress} [Mem. Am. Math. Soc. 708, 168 p. (2001; Zbl 1053.20045)] and comparable with a similar algorithm for SL due to \textit{M. Conder} and \textit{C. R. Leedham-Green} [Groups and computation III. Proceedings of the international conference at the Ohio State University, Columbus, OH, USA, June 15-19, 1999. Berlin: Walter de Gruyter. Ohio State Univ. Math. Res. Inst. Publ. 8, 113-121 (2001; Zbl 1012.20043)]. Also concrete runtimes from the author's implementation are given.
reviewed by: A. Hulpke
Mathematics Subject Classification ID: 20-04
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 68W30
Mathematics Subject Classification ID: 20C40
Mathematics Subject Classification ID: 20P05
Mathematics Subject Classification ID: 68W20
zbMATH DE Number: 2056560
zbMATH Keywords: matrix groups
zbMATH Keywords: simple groups
zbMATH Keywords: constructive recognition
zbMATH Keywords: Las Vegas algorithms
describes a project that uses: GAP
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2056560